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Characterization of the space of solutions of the membrane shell equation for arbitrary C[sup 1'1] midsurfaces

Delfour M.

Control and Cybernetics

1999

Vol. 28, no 3

481-501

A comprehensive convergence theory of dynamical thin shell models for the purposes of control theory relies heavily on a thorough analysis of the static model and the complete specification of the spaces of solutions of the asymptotic solution for general midsurfaces ranging from the plate to arbitrary C[sup 1,1] midsurfaces. In this paper the existence of solution to the membrane shell equation is studied in a bounded open connected domain [omega] (Lipschitzian when [omega] has a boundary [gamma]) in a C[sup 1,1] midsurface for homogeneous Neumann boundary conditions or homogeneous Dirichlet boundary conditions on a part [gamma,omikron] of [gamma]. It is proved that its tangential part is solution of the reduced membrane shell equation in H[sup 1](omega)[sup N] (resp. H[sup 1][sub gamma,omikron](omega)[sup N]) unique up to an element of a finite dimensional subspace, while its normal component belongs to a weighted L[sup 2](omega) space by the pointwise norm of the second fundamental form. It is also shown that the reduced equation is equivalent to the equation for the projection onto the linear subspace of vector functions whose linear change of metric tensor is orthogonal to the second fundamental form of the midsurface.